#4308129

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

5285 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)(15,17)(16,19)(18,20)(23,27)(25,28);(6,9)(10,12)(14,15)(18,19)(21,22,24,26)(23,25,27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,22)(23,25)(24,26)(27,28);(21,23)(22,25)(24,27)(26,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,25)(24,27)(26,28)

Polycyclic presentation:

2^9|b^2=i,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326256, #72326262, #72326273
maximal normal subgroup of	#72326256, #72326262, #72326273
maximal quotient of	#72328394, #72330281

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215325);