#7627033

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×#3552	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6836 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)(6,9)(10,12)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26);(1,2,4,3)(14,15)(19,21)(20,22)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,21,24,19)(17,22,25,20)(23,27,28,26);(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27);(5,6)(7,10)(8,9)(11,12)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=h,e^2=i,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, 7534229);