#7626880

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×#3518	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6834 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)(13,14,18,21)(15,19,23,17)(16,24,26,20)(22,25,28,27);(13,15,18,23)(14,17,21,19)(16,22,26,28)(20,25,24,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,21)(18,23)(20,27)(24,25)(26,28);(5,7,8,11)(6,10,9,12)(13,16,18,26)(14,20,21,24)(15,22,23,28)(17,25,19,27);(1,3,4,2)(5,7)(6,10)(8,11)(9,12);(5,6)(7,10)(8,9)(11,12)(13,17,18,19)(14,15,21,23)(16,25,26,27)(20,22,24,28)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=gh,e^2=gi,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, 7534076);