#392135

order	512
nilpotency class	3
isoclinism	isoclinic class of #392135
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14679
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×H(7,𝔽₂)	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

16105 conjugacy classes of subgroups; only the 688 normal subgroups are shown.

normal subgroups only

This group has 16105 conjugacy classes of subgroups — too many to draw. Only its 688 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,11,14,7)(4,15,17,8)(5,9)(6,10)(12,19)(13,20,27,24)(16,28,29,21)(18,22)(23,26)(25,31)(30,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,20)(10,11)(15,23)(16,25)(17,26)(18,27)(21,30)(22,24)(28,31)(29,32);(1,4)(2,8)(3,12)(5,16)(6,17)(7,19)(9,21)(10,15)(11,23)(13,25)(14,26)(18,29)(20,30)(22,28)(24,31)(27,32);(1,5)(2,9)(3,13)(4,16)(6,18)(7,20)(8,21)(10,22)(11,24)(12,25)(14,27)(15,28)(17,29)(19,30)(23,31)(26,32)

Polycyclic presentation:

2^9|a^2=h,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^b=gi,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72241398, #72252410, #72252520, …
maximal subgroup of	#72210025, #72210026, #72210035, …
maximal normal subgroup of	#72210025, #72210026, #72210035, …
maximal quotient of	#72225809, #72225822, #72225826, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299331);