#6545110

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2077 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452306);