#508565

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fi,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, #72343823, #72343824
maximal normal subgroup of	#72343796, #72343823, #72343824

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415761);