#392234

order	512
nilpotency class	3
isoclinism	isoclinic class of #190022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3434 conjugacy classes of subgroups; only the 434 normal subgroups are shown.

normal subgroups only

This group has 3434 conjugacy classes of subgroups — too many to draw. Only its 434 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,10,16,11,15,9,14)(7,17,19,30,20,12,25,27)(8,21,24,28,23,13,22,29)(18,26,32,35,33,31,34,36);(5,7)(6,12)(8,18)(9,19)(10,25)(11,20)(13,26)(14,27)(15,17)(16,30)(21,31)(22,32)(23,33)(24,34)(28,35)(29,36);(5,8)(6,13,15,21)(7,18)(9,22)(10,24)(11,23)(12,26,17,31)(14,28,16,29)(19,32)(20,33)(25,34)(27,35,30,36);(2,4)(5,9,11,10)(6,14,15,16)(7,19,20,25)(8,22,23,24)(12,27,17,30)(13,28,21,29)(18,32,33,34)(26,35,31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317836, #72318269, #72318323
maximal normal subgroup of	#72317836, #72318269, #72318323

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299430);