#508643

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,d^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343796, #72343837, #72343838, …
maximal normal subgroup of	#72343796, #72343837, #72343838, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415839);