#392195

order	512
nilpotency class	3
isoclinism	isoclinic class of #32947
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3392 conjugacy classes of subgroups; only the 498 normal subgroups are shown.

normal subgroups only

This group has 3392 conjugacy classes of subgroups — too many to draw. Only its 498 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341657, #72341755, #72341764, …
maximal subgroup of	#72281136, #72281182, #72281258, …
maximal normal subgroup of	#72281136, #72281182, #72281258, …
maximal quotient of	#72287575, #72313299

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299391);