#405607

order	512
nilpotency class	3
isoclinism	isoclinic class of #394273
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1531 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251368, #72251390, #72251399, …
maximal normal subgroup of	#72251368, #72251390, #72251399, …
maximal quotient of	#72251460, #72268301

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312803);