#610025

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3972 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303658, #72303664, #72303670, …
maximal normal subgroup of	#72303658, #72303664, #72303670, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 517221);