#7199164

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#45042	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1886 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=gi,d^2=fhi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7106360);