#459408

order	512
nilpotency class	3
isoclinism	isoclinic class of #456510
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1173 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366604);