#7989741

order	512
nilpotency class	2
isoclinism	isoclinic class of #7989741
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

7534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336756
maximal normal subgroup of	#72336756
maximal quotient of	#72336807, #72336810, #72336858

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7896937);