#7626577

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×D₄×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533773);