#146307

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,4)(9,10,12,17,13,19,24,32,14,18,23,33,25,16,29,31)(11,15,21,27,22,26,36,37,20,28,34,39,35,30,40,38);(2,4)(5,6,7,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,30)(17,27)(18,28)(19,26)(23,34)(24,36)(25,35)(29,40)(31,38)(32,37)(33,39);(10,16,18,19)(11,20)(12,23)(15,26,28,30)(17,32,33,31)(22,35)(24,29)(27,38,39,37)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=h,h^2=i,b^a=bd,c^a=ch,g^a=gi,c^b=ci,e^c=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319819, #72319826, #72319850, …
maximal normal subgroup of	#72319819, #72319826, #72319850, …
maximal quotient of	#72322457

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53503);