#544315

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁴×C₄³

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	C₂×#2388	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3222 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=f,e^2=h,b^a=bf,c^a=cg,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303956, #72303966, #72304096
maximal normal subgroup of	#72303956, #72303966, #72304096

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 451511);