#7504112

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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1885 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)(13,14,18,23,19,24,17,28)(15,20,25,22,16,26,27,21);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,16)(14,20,24,26)(17,27,18,25)(21,23,22,28);(2,4)(13,15,19,16)(14,21,24,22)(17,25,18,27)(20,23,26,28);(6,11)(7,12)(13,16,19,15)(14,22,24,21)(17,27,18,25)(20,28,26,23);(5,8,9,10)(6,7,11,12)(13,17,19,18)(14,23,24,28)(15,25,16,27)(20,21,26,22)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,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, 7411308);