#457277

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1184 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=i,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=de,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364473);