#385699

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

840 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,27,22,26,21,25)(19,20,29,32,28,31,24,30);(1,3,5,2)(4,7,8,6)(10,13)(11,15)(17,19,22,28)(18,20,26,31)(21,24,23,29)(25,30,27,32);(1,4,5,8)(2,6,3,7)(10,13)(14,16)(17,20,22,31)(18,24,26,29)(19,25,28,27)(21,30,23,32);(2,3)(6,7)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,25,26,27)(19,24,28,29)(20,30,31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ef,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292895);