#184809

order	512
nilpotency class	3
isoclinism	isoclinic class of #184805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92005);