#7626832

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×#3460	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7276 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=gh,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, 7534028);