#7656994

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

7070 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)(5,6,11,10)(7,15,18,12)(8,13,17,19)(9,14,16,20)(24,32)(27,34)(31,35)(33,36);(1,2)(3,4)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,34)(33,35);(5,7)(6,12)(8,16)(9,17)(10,15)(11,18)(13,14)(19,20)(21,23,26,28)(22,25,29,30)(24,31,32,35)(27,33,34,36);(5,8,11,17)(6,13,10,19)(7,16,18,9)(12,14,15,20)(21,24)(22,27)(23,31)(25,33)(26,32)(28,35)(29,34)(30,36);(5,9,11,16)(6,14,10,20)(7,17,18,8)(12,13,15,19)(21,22)(23,25)(24,27)(26,29)(28,30)(31,33)(32,34)(35,36);(5,10,11,6)(7,15,18,12)(8,19,17,13)(9,14,16,20)(21,25)(22,28)(23,29)(24,33)(26,30)(27,35)(31,34)(32,36)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283309, #72283676
maximal normal subgroup of	#72283309, #72283676

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564190);