#514515

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂³	C₂⁴×C₄
Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304056, #72304075
maximal normal subgroup of	#72304056, #72304075

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 421711);