#7899249

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6923 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)(5,6,10,7)(8,13,16,11)(9,12,17,14)(15,19,20,18)(21,22)(23,26)(24,29)(25,28)(27,32)(30,34)(31,36)(33,35);(1,2,3,4)(21,22,26,23)(24,27,32,29)(25,28,34,30)(31,35,33,36);(6,7)(11,13)(12,14)(18,19)(22,23)(27,29)(28,30)(35,36);(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,19,20,18)(21,23,26,22)(24,29,32,27)(25,30,34,28)(31,36,33,35);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,24,26,32)(22,27,23,29)(25,33,34,31)(28,36,30,35);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,25,26,34)(22,28,23,30)(24,31,32,33)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303354
maximal normal subgroup of	#72303354

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7806445);