#158780

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

3292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fi,d^2=fgh,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227640
Fitting subgroup of	#72227640, #72227659
maximal subgroup of	#72227640, #72227650, #72267004, …
maximal normal subgroup of	#72227640, #72227650, #72267004, …
maximal quotient of	#72227638

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 65976);