#7626695

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×#3520	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7010 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)(6,9)(10,12)(13,14,18,19)(15,16,21,24)(17,20,25,26)(22,23,27,28);(1,3,4,2)(14,19)(15,21)(20,26)(22,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,22)(18,21)(19,24)(20,23)(25,27)(26,28);(7,11)(10,12)(13,16)(14,15)(17,23)(18,24)(19,21)(20,22)(25,28)(26,27);(5,7,8,11)(6,10,9,12);(5,6)(7,10)(8,9)(11,12)(13,17,18,25)(14,20,19,26)(15,22,21,27)(16,23,24,28)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,e^2=h,f^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, 7533891);