#438981

order	512
nilpotency class	3
isoclinism	isoclinic class of #422540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285161, #72285454, #72285492, …
maximal normal subgroup of	#72285161, #72285454, #72285492, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346177);