#93648

order	512
nilpotency class	4
isoclinism	isoclinic class of #93523
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×C₄×Q₈	C₂²
prime radical	C₂⁴	#95
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄×Q₈ ▷ C₂⁴ ▷ C₂ ▷ C₁

377 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=fhi,e^2=fhi,b^a=bc,c^a=cf,e^a=efgh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72266637, #72266638
Fitting subgroup of	#72266637
maximal subgroup of	#72266598, #72266637
maximal normal subgroup of	#72266598, #72266637
maximal quotient of	#72267637

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 844);