#7626573

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	C₂²×#548	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8200 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)(13,14,19,16)(15,22,18,20)(17,24,26,21)(23,27,25,28);(13,15,19,18)(14,20,16,22)(17,23,26,25)(21,27,24,28);(5,6)(7,10)(8,9)(11,12)(13,16,19,14)(15,20,18,22)(17,24,26,21)(23,27,25,28);(7,11)(10,12)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28);(5,7)(6,10)(8,11)(9,12)(13,18,19,15)(14,20,16,22)(17,25,26,23)(21,27,24,28);(5,6)(7,10)(8,9)(11,12)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,e^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, 7533769);