#394065

order	512
nilpotency class	3
isoclinism	isoclinic class of #394064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4207 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|c^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236586, #72238117, #72251134, …
maximal normal subgroup of	#72236586, #72238117, #72251134, …
maximal quotient of	#72236728, #72238118, #72238120, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301261);