#7626828

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×#537	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=h,e^2=h,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, 7534024);