#7185378

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
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×#3330	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1924 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)(6,10)(8,11)(13,14,18,22)(15,23,26,27)(16,21,17,20)(19,25,28,24);(5,6,9,10)(7,8,12,11)(13,15)(14,19)(16,24)(17,25)(18,26)(20,23)(21,27)(22,28);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(13,16,18,17)(14,20,22,21)(15,24,26,25)(19,23,28,27);(6,10)(7,12)(15,25)(16,17)(19,27)(20,21)(23,28)(24,26);(5,8,9,11)(6,7,10,12)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,f^d=fi

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, 7092574);