#154678

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278086, #72278103, #72278119, …
maximal normal subgroup of	#72278086, #72278103, #72278119, …
maximal quotient of	C₂×#154678, #72299914

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61874);