#774649

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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

3298 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)(5,6)(7,12)(8,11)(9,10)(13,14)(15,19)(16,20)(17,18)(21,22,25,27)(23,26,28,24);(6,10)(11,12)(14,18)(15,20)(21,23,25,28)(22,24,27,26);(5,7,9,8)(6,11,10,12)(13,15,17,20)(14,19,18,16)(21,24,25,26)(22,23,27,28);(1,2)(3,4)(5,8,9,7)(6,11,10,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,27)(26,28);(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=f,d^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326293, #72326297, #72326373
maximal normal subgroup of	#72326293, #72326297, #72326373
maximal quotient of	#72333107

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 681845);