#418136

order	512
nilpotency class	3
isoclinism	isoclinic class of #417508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#2539	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2443 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262061, #72262082
maximal normal subgroup of	#72262061, #72262082

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325332);