#376782

order	512
nilpotency class	3
isoclinism	isoclinic class of #24420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14185
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213889, #72213913, #72219494
maximal normal subgroup of	#72213889, #72213913, #72219494

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283978);