#6380975

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2031 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250423, #72256929, #72257050
maximal normal subgroup of	#72250423, #72256929, #72257050
maximal quotient of	#72264604, #72308242

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288171);