#7622796

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁵
commutator, socle	C₂²	C₂⁴×C₈
prime radical	#3482	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,27,23,24,35,37,21,25,34,39,19,28,30,42)(20,26,33,41,31,36,44,46,32,38,45,47,29,40,43,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,23)(18,24,25,28)(20,31,32,29)(22,30,34,35)(26,40,38,36)(27,37,39,42)(33,44,45,43)(41,48,47,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,23)(18,24,25,28)(20,29,32,31)(22,30,34,35)(26,36,38,40)(27,37,39,42)(33,43,45,44)(41,46,47,48);(2,8)(6,9)(7,16)(14,15)(18,25)(24,28)(26,38)(27,39)(36,40)(37,42)(41,47)(46,48);(17,20,22,33,23,31,35,44,21,32,34,45,19,29,30,43)(18,26,27,41,24,36,37,46,25,38,39,47,28,40,42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,h^2=i,b^a=bf,c^a=ci,d^a=df,c^b=cfi,e^b=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287764, #72320055, #72320093, …
maximal normal subgroup of	#72287764, #72320055, #72320093, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529992);