#7626131

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator	C₂²	C₂⁵×C₄
Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3480	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7182 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)(5,6)(7,11)(8,10)(9,12)(13,14)(15,21)(16,20)(17,19)(18,22)(23,27)(24,26)(25,28);(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,23)(17,22)(19,21)(20,25)(24,28)(26,27);(5,7,9,8)(6,10,12,11)(15,22)(18,21)(23,28)(25,27);(1,3,4,2)(5,7,9,8)(6,11,12,10)(14,19)(18,21)(20,26)(25,27);(5,8,9,7)(6,11,12,10);(13,16,17,24)(14,20,19,26)(15,23,22,28)(18,25,21,27)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=cg,d^b=dh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533327);