#479897

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈² · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄×C₈	C₂⁴
commutator	C₂	C₂²×C₈²
socle	C₂²	C₂²×C₄×C₈
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=g,f^2=h,g^2=i,d^a=di,c^b=ci

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224814, #72224817, #72224818, …
maximal normal subgroup of	#72224814, #72224817, #72224818, …
maximal quotient of	#72226965, M₃₂²

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387093);