#7502969

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#30505	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1901 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^b=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410165);