#7641797

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3475	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

(2,4)(5,6)(7,9)(8,16)(10,13)(11,14)(12,18)(15,19)(17,20)(21,22)(23,26)(24,31)(25,27)(28,32)(29,34)(30,36)(33,35);(1,2,3,4)(5,7,11,13)(6,9,14,10)(8,15,18,20)(12,17,16,19)(21,23,25,29)(22,26,27,34)(24,30,32,35)(28,33,31,36);(5,8)(6,12)(7,15)(9,17)(10,19)(11,18)(13,20)(14,16)(22,27)(23,29)(28,31)(30,35);(5,9,11,10)(6,13,14,7)(8,17,18,19)(12,20,16,15)(21,24,25,32)(22,28,27,31)(23,30,29,35)(26,33,34,36);(8,18)(12,16)(15,20)(17,19);(5,10,11,9)(6,13,14,7)(8,19,18,17)(12,20,16,15)

Polycyclic presentation:

2^9|b^2=gh,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7548993);