#377325

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=efi,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284521);