#362127

order	512
nilpotency class	3
isoclinism	isoclinic class of #24808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1236 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,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, 269323);