#903713

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316580, #72316591, #72316597
maximal normal subgroup of	#72316580, #72316591, #72316597

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 810909);