#376781

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, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1275 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6,12,16)(7,17,21,22)(8,10,15,11)(9,24,26,23)(13,20,29,19)(14,27,30,25)(18,28,33,36)(31,34,35,32);(5,7,12,21)(6,13,16,29)(8,22,15,17)(9,18,26,33)(10,20,11,19)(14,28,30,36)(23,35,24,31)(25,34,27,32);(5,8,11,6,12,15,10,16)(7,17,20,13,21,22,19,29)(9,23,25,14,26,24,27,30)(18,31,32,28,33,35,34,36);(1,3,4,2)(5,9,12,26)(6,14,16,30)(7,18,21,33)(8,23,15,24)(10,27,11,25)(13,28,29,36)(17,31,22,35)(19,34,20,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=h,e^2=i,f^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283977);