#6371507

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342612
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#3487	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

6254 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306319, #72306520, #72317836, …
maximal normal subgroup of	#72306319, #72306520, #72317836, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278703);