#363951

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3494	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fi,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318100, #72318108, #72318124
maximal normal subgroup of	#72318100, #72318108, #72318124

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271147);