#184752

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,g^b=gh,d^c=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91948);