#795842

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	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3642 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)(7,11)(8,12)(9,10)(14,18)(15,20)(21,22)(23,26)(24,27)(25,28);(6,10)(8,11)(13,14,17,18)(15,19,20,16)(21,23,24,28)(22,25,27,26);(1,2)(3,4)(21,22)(23,26)(24,27)(25,28);(5,7)(6,11)(8,10)(9,12)(13,15)(14,19)(16,18)(17,20)(21,22)(23,25)(24,27)(26,28);(5,8,9,11)(6,12,10,7)(13,16,17,19)(14,20,18,15)(21,23)(22,26)(24,28)(25,27)

Polycyclic presentation:

2^9|b^2=hi,e^2=fh,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	#72232459, #72232471, #72232545, …
maximal normal subgroup of	#72232459, #72232471, #72232545, …
maximal quotient of	#72330943

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 703038);