#412175

order	512
nilpotency class	3
isoclinism	isoclinic class of #395246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1606 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319371);