#7629679

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8396 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,25)(17,21)(18,22)(19,23)(24,28)(26,27);(6,9)(10,12)(13,15,19,18)(14,20,23,22)(16,26,17,24)(21,28,25,27);(2,4)(13,16,19,17)(14,21,23,25)(15,24,18,26)(20,27,22,28);(13,17,19,16)(14,21,23,25)(15,24,18,26)(20,28,22,27);(13,18,19,15)(14,22,23,20)(16,26,17,24)(21,28,25,27);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,c^b=ci,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, 7536875);