#6374041

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2428	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2053 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)(8,9)(11,12)(13,14,19,21,18,23,16,22)(15,24,26,28,27,20,25,17);(6,9)(10,12)(13,15)(14,20)(16,25)(17,21)(18,27)(19,26)(22,28)(23,24);(1,2,3,4)(13,16,18,19)(14,21,23,22)(15,25,27,26)(17,24,28,20);(13,17,19,24,18,28,16,20)(14,15,21,26,23,27,22,25);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,22,23,21)(15,26,27,25)(17,24,28,20)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281237);